Spectral Properties of Non-Stationary Systems of Linear Stochastic Difference Equations

A method is proposed to eliminate trends from a sample from a non-stationary system of linear stochastic difference equations. The auto-covariance matrix and the spectral density matrix of the detrended component of the sample are derived. The latter matrix turns out to have the same form as the spectral density matrix for a stationary system when expressed in terms of the roots of the system. Since the parameters of the system are assumed known throughout the paper, the problem of statistical inference does not arise.